If ` sin theta = cos theta` ,then the value of `2 tan^(2) theta + sin^(2) theta -1` is equal to

Options

A. `1/2`
B. `5/2`
C. `3/2`
D. 2

Explanation

To solve the problem, we start with the given equation:

**Step 1: Understand the equation**
Given that \( \sin \theta = \cos \theta \).

**Step 2: Find the relationship between sine and cosine**
From the equation \( \sin \theta = \cos \theta \), we can divide both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \frac{\sin \theta}{\cos \theta} = 1 \]
This simplifies to:
\[ \tan \theta = 1 \]

**Step 3: Determine the angle**
The angle \( \theta \) for which \( \tan \theta = 1 \) is:
\[ \theta = 45^\circ \quad \text{(or } \frac{\pi}{4} \text{ radians)} \]

**Step 4: Calculate \( \tan^2 \theta \) and \( \sin^2 \theta \)**
Now, we can find \( \tan^2 \theta \) and \( \sin^2 \theta \):
\[ \tan^2 45^\circ = 1^2 = 1 \]
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

**Step 5: Substitute into the expression**
Now we substitute these values into the expression \( 2 \tan^2 \theta + \sin^2 \theta - 1 \):
\[ 2 \tan^2 \theta + \sin^2 \theta - 1 = 2(1) + \frac{1}{2} - 1 \]

**Step 6: Simplify the expression**
Calculating this step-by-step:
\[ = 2 + \frac{1}{2} - 1 \]
\[ = 2 - 1 + \frac{1}{2} \]
\[ = 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]

**Final Answer:**
Thus, the value of \( 2 \tan^2 \theta + \sin^2 \theta - 1 \) is:
\[ \frac{3}{2} \]

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